C₅×#5141 #11750640

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#5141

Splittings

Fitting	C₃×C₅×#131 ⋊ C₂
Hall	C₃×C₅ ⋊ #471
Sylow	C₃ ⋊ C₅×#471

Subgroups

	subgroup	G/H
center	C₂×C₅	#930
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#131	C₂
socle	C₂×C₃×C₅	#121
prime radical	C₅×#949	C₂
hypercenter	C₅×#131	D₃
hypocenter	C₃	C₅×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#131

208 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,4,5)(6,7,8)(9,11)(10,15)(12,18)(13,19)(14,21)(16,24)(17,20)(22,23);(9,10,12,14,13,16,20,22)(11,15,18,21,19,24,17,23);(7,8)(10,14)(11,17)(12,20)(16,22)(18,19)(21,23)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#471, C₂×C₅×#5141, D₅×#5141, …
maximal normal subgroup of	C₅×D₃×#471, C₂×C₅×#5141, D₅×#5141, …
maximal quotient of	C₅×#70846, C₅×#70938, C₅×#70851, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4835);